Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem. I would love some blogger critique so please let me know what you think...

HORIZON: How far away is the horizon?

This is a pretty famous problem in mathematics (I think). I remember seeing it and being intrigued. I think there is an opportunity for pretty rich math here. Students would have to create an abstraction to represent the situation, develop rules about their abstraction, and use them to approximate an answer to the unit question.

PROBLEMATIC PACKAGING: How can you optimize this packaging?

This one arose out of a curiosity I had about optimizing parking lot design. I thought it might be fun as a puzzle where students get a package and different "items" (blocks of two or three sizes and shapes and worth various points) and they have to figure out how to maximize their point value. Could be extended by looking at different point values or different size boxes.

GET ME OUTTA HERE: How do you know if a game is solvable?

I love these puzzles. I'm not exactly sure how to turn this into a full-blown unit, but I'm pretty sure it can be done. This might not be the best unit question. I have also considered giving students a specific puzzle and asking "What is the fewest number of moves?" Maybe we can extend it from there and move into "solvable" setups?

7 CLICKS FROM KEVIN BACON: What is the minimum number of clicks to get to Kevin Bacon from ANY person on Facebook?

This one I'm REALLY not sure about. Obviously, it comes from the popular game about social connection (although it might be a good idea to use a celebrity my student will have actually hear of). I thought there might be some connections to graph theory here and it might be a nice extension of a combinatorics unit that I will do again next year ("How many combinations are there at Chipotle?"). Feedback please.

WIN, LOSE, or DRAW: Can you draw this without lifting your pencil or going over the same line twice?

The Bridges of Konigsberg is such a GREAT problem that I want to turn it into a whole unit next year. I'm not sure I love the context of the original problem because, to students, I think it feels like a pseudocontext. I thought about giving them a crazy network diagram and asking the unit question about that (maybe with "bridges" showing up along the way?). However, I usually prefer to start with a concrete situation and abstract from there....not sure.

...something more complicated but this was the best picture I could find for now

GOING COASTAL!: How long is the California coastline?

I did a similar project this year where students figured out the area of the Koch Snowflake. I'm thinking about trying to start concrete and move abstract next year with this one. I'm sure the Koch Snowflake will still rear it's ugly head somewhere in our investigation.

I have thought of the horizon as a problem, not really a project, but here's my 101qs submission for a starter on that http://www.101qs.com/411-stairway-to-heaven. The tricky part about horizon that I discovered when I researched the solution to this problem is refraction due to the atmosphere. I am uncertain about using it because it would be pretty intense to have students find the refraction of the earth's atmosphere within a math class (as opposed to a physics class). I don't know enough about refraction to know if this is doable.

I love the bridges of Konisburg as well, and I have always wondered if we could turn it into an exploration of board games that have connections (I am thinking Ticket to Ride and Pandemic as my faves). I don't know though.

I am all about anything with fractals. I love them; kids love drawing them. I'd love to see what you can do with this.

@Timon Thanks for the heads up about refraction! I'll have to look into that. On a purely mathematical basis, I think there is still a lot of rich-ness here. I'm also going to check out the games you recommended.

I love the idea of project based learning, and your proposals. I'm headed in that direction too, but it'll take some convincing of others. I especially like the packaging project. Students can physically have the item in front of them. After some research, I'd suspect that most companies have loads of reasons why their products are packaged as they are. Perhaps, students can submit a better design. Maximizing one quantity while minimizing another is an important concept.
California coastline-cool idea. I'd imagine that students would be curious. Recently our district redefined the school boundaries and provided what I think is a gold mine, an irregularly shaped map drawn to scale. I'd like to have my students calculate how many square miles our school's new boundaries capture, although the hook question needs to be crafted better.
Here's an idea I'm trying to develop. In La Mesa the city has installed parking meters that zero out the credit once the vehicle leaves the spot. If 12 minutes are left and the vehicle backs out, the time goes to zero. Not cool. But it has me wondering how much income one parking meter could generate on a "good" day with such a fee structure.
Keep up the good work at HTH!